2. Problem 4.38 There are two machines available for cutting corks intended for

Question

2. Problem 4.38 There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 cm and standard deviation.1 cm. The second machine produces corks with diameters that have a normal distribution with mean 3.04 cm and standard deviation.02 cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm. (a) Which machine is more likely to produce an acceptable cork? Why? (b) What is the defects per million opportunities (DPMO) for each machine? (c) In what range will 95% of forks produced by each machine lie

Explanation / Answer

(a)

In this question we have
Machine 1:
X ~ Normal( x = 3 , x² = 0.01 )
X ~ Normal( x = 3 , x = 0.1 )

Find P( 2.9 < X < 3.1 )
= P( ( 2.9 - 3 ) / 0.1 < ( X - ) / < ( 3.1 - 3 ) / 0.1 )
= P( -1 < Z < 1 )
= P( Z < 1 ) - P( Z < -1 )
= 0.8413447 - 0.1586553
= 0.6826895

Machine 2:
X ~ Normal( x = 3.05 , x² = 0.0064 )
X ~ Normal( x = 3.05 , x = 0.08 )

Find P( 2.9 < X < 3.1 )
= P( ( 2.9 - 3.05 ) / 0.08 < ( X - ) / < ( 3.1 - 3.05 ) / 0.08 )
= P( -1.875 < Z < 0.625 )
= P( Z < 0.625 ) - P( Z < -1.875 )
= 0.7340145 - 0.03039636
= 0.7036181

Machine 2 is more likely to produce an acceptable cork.

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